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UC-NRLF 


SB    531    MT? 


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LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 

GIFT     OK 

Received  O^&A  ,  ,  i8g  /. 

A  ccessions  No  ^c^t&p3    Shelf  No. ... 

<,£♦ *3v> 


KEY 


ELEMENTARY      TREATISE 


ON 


ALGEBRA 


BY     THOMAS    SHERWIN,     A.M 

Principal  of  the  E.iglish  High  School,  Boston. 


tfUj*  OF  THB 


f>  *         D* 


.     '>: 


BOSTON 


NICHOLS     &    HALL, 

32  Bhomfield  Skteet. 


O  3  * 


Entered  according  to  act  of  Congress,  in  the  year  1842, 

By  Thomas  Sherwin, 

in  the  Clerk's  Office  of  the  District  Court  of  Massachusetts. 


^  v 


f/J 


JHI7BESIT7; 


KEY. 


[  In  the  first  edition  of  the  Algebra,  the  Articles  are  numbered  erroneously  from  the  com- 
mencement  of  Section  XXVI.  To  make  that  edition  correspond  with  the  Key,  increase  by 
2  the  number  of  each  Article,  commencing  with  the  Section  specified.] 


SECTION  I. 

Art.  4. 

1.  Performed. 

2.  Horse  $100.     Saddle  $10. 

3.  6  bushels  of  each. 

4.  11  of  each. 

5.  5  melons ;  10  pears ;  30  peaches. 

6.  15  days.     Man  $20;  boy  $10. 

7.  105,  210,  and  315. 

8.  25,  35,  and  40. 

9.  10  hours.     One  goes  80  miles,  the  other  70  miles. 

10.  1st  8,  2d  16,  3d  24  guineas. 

11.  Wife  $36000;  elder  son  $27000;  younger  son  $18000. 

12.  120  cavalry  ;  360  artillerymen ;  720  infantry. 

13.  A  8  days,  72  rods;  B  16  days,  112  rods;  C  80  days,  400 
rods. 

14.  8  yds,  16  yds,  and  24  yds  respectively. 

15.  Sheep  $2;  calf  $4;  cow  $20. 

16.  Greater  14  galls;  less  7  galls. 

17.  Beef  10  lbs;  mutton  20  lbs. 

18.  32  of  each. 

19.  80  guineas ;  240  crowns. 

20.  10,  20,  30,  60  and  30  miles,  on  the  respective  days. 


KEY.  IT 


21.  8  bushels  of  each. 

22.  Cows  $25,  oxen  $50  each. 


SECTION   fl. 

Art.  7. 

1.  Performed. 

2.  1st  $23000;  2d  $33000. 

3.  73£  and  156f  miles. 

4.  A's  15,  B's  35,  C's  50  years. 

5.  1st  8  lbs ;  2d  22  lbs ;  3d  30  lbs ;  4th  60  lbs. 

6.  6,  8,  10,  12,  and  14  years  respectively. 

7.  Black  $30;  green  $32;  blue  $35. 

8.  A's  30,  B's  20  years. 

9.  Each  daughter  $1000  ;    younger  son  $2000  ;    elder  son 
$5000. 

10.  Performed. 

Art.  9. 

1.  493  and  40o  votes  respectively. 

2.  5  men ;  20  boys ;  180  girls. 

3.  150  cavalry;  450  riflemen ;   1700  infantry. 

4.  A  $2750;  B  $8250;  C  $22000;  D  $32000. 

5.  A  $377;  B  $7;  C$107. 

6.  10  years  in  France;  60  in  England;  10  in  America. 

7.  1500  English ;  900  Irish ;  500  French. 

8.  50  eagles;  90  half-eagles ;  250  dollars. 

9.  Broadcloth  50 ;  cassimere  144 ;  silk  288  yards. 

10.  $2000. 

11.  A  $10000;  B  $10600;  C$20800;  D  $20300;  E  $30800 

12.  80  oxen;  215  cows;  215  calves;  395  sheep. 

13.  20  students;  55  merchants;  65  officers. 


Ill,  IV  KEY. 


SECTION    III. 

Art.  11. 

1.  Performed. 

2.  Finer  $6 ;  coarser  $4  per  yard. 

3.  A's  50 ;  B's  40  years. 

4.  20  each. 

5.  Sum  distributed  153d.  Eldest  60d. ;  2d  42d. ;  3d  30d. ;  4tl 
21c5 

3.  1st  10s. ;  2d  7s. ;  3d  8s. ;  4th  5s. ;  5th  3s. ;  6th  2s. 

7.  Smaller  40  galls. ;  larger  80  galls. 

8.  Wheat  8s. ;  rye  6s.  per  bushel. 

9.  45  scholars  in  all.     He  wished  to  place  7  in  a  row. 

10.  10  lbs;  95  cents. 

11.  10  shillings. 

12.  $60. 

13.  22  years. 

14.  12  years  ago. 

15.  150  sheep. 

16.  8  days ;  720  miles  from  the  starting  place. 

17.  Horse  $200;  chaise  $275. 

18.  Wheat  8s. ;  oats  3s. 

19.  $1000. 

20.  125  miles. 


SECTION    IV. 

Art.  IS. 

1st,  2d,  3d,  performed. 

4.  60  years  old.     20  years  in  Germany ;  15  in  France  ;  24  in 
England. 

5.  Whole  estate  $5040.     He  owed  his  creditors  respectively 
$630,  $720,  and  $672. 

I* 


6  KEY.  V. 

6.  Whole  mixture  32  oz.     12§oz.  gold;  16  oz.  silver;  3£  oz. 
copper. 

7.  Whole  estate  $22500.    Wife's  portion  $6500 ;  eldest  son's 
$6500 ;  2d  son's  $5750 ;  youngest  son's  $3750. 

8.  80  years. 

9.  A's  40 ;  B's  30  years. 
10  30  inches. 

11.  72. 

12.  $150. 

13.  $75. 

14.  1st  30  cents ;  2d  and  3d  15  cents  each. 

15.  Whole  number  30.     The  1st  had  10;  2d  6;  3d  4;  4th  10 

16.  1st  horse  $100;  2d  horse  $150;  chaise  $300. 

17.  Income  $1200 ;  he  spends  $650. 

18.  45  sheep. 

19.  1st  £56;  2d  .£35;  3d  £7. 

20.  280  miles  in  all.    40  miles  1st  day;  56  2d  day;  70  3d  day. 


SECTION    V. 

Art.  13. 

1st,  2d,  3d,  performed. 

4.  A  $300 ;  B  $200. 

5.  Man  40 ;  woman  30  years. 

6.  Man's  age  50  ;  woman's  45  years 

7.  £50. 

8.  From  A  to  B  40  miles ,  from  B  to  C  60  miles ;  from  A  to 
C  100  miles. 

9.  1st  105  yds;  2d  120  yds;  3d  135  yds. 

10.  Each  has  £150. 

11.  100  yards. 

12.  Each  has  $140. 

13.  A  $350 ;  B  $400 ;  C  $475. 

14.  30  cows ;  69  sheep. 


VI,  VII.  KEY. 

15.  840. 

16.  $1000. 

1 7.  Cider  $4 ;  beer  $6  per  barrel. 

18.  54  and  60. 

19.  Shorter  20  rods;  longer  40  rods. 

20.  Chaise  $233£;  horse  $166§. 

21.  $100. 

22.  1st  car  18  miles;  2d  car  20  miles  per  hour. 

23.  $150. 

24.  $110. 

25.  139. 


SECTION    VI. 

Art.  15. 

1.  Horse  $150;  chaise  $250. 

2.  80  gallons. 

3.  Greater  35  ;  less  25. 

4.  Performed. 

5.  20  tons. 

6.  $72. 

7.  Greater  75;  less  72. 

8.  100  gallons  each. 

9.  90  eggs. 

10.  Time  past  40  years. 

11.  Lower,  middle  and  upper  parts  respectively,  12,  16  and  20 
feet.    Whole  pole,  48  feet. 

12.  20. 


v  SECTION    VII. 

Art.  17. 

1.  liabcm.  3.  2labxy. 

2.  845c.  4.  Ziapqr. 


KEY.  VIII,  IX. 


5.  Slaghx  9.  21  mqs. 

6.  156  a  x  y.  10.  OOaqx. 

7.  Amnpq.  11.  Performed. 

8.  20 amp. 

Art.  19. 

1.  Performed.  11.  12  a?  h6n*x7 

2.  54<z62c.  12.  567a3r2*7y. 

3.  301  a4  66.  13.  \Z2p3xSy3. 

4.  46  a1  m7  x  14.  36  a3  m5  n-pP 

5.  399  a9  7?2  x.  15.  39z3#3z4. 

6.  llx*y8.  16.   170a562cz. 

7.  72  a3  b3c3(P.  17.  210a563c. 

8.  146  tt*9  ti5x4.  18.  54c2^z4y3. 

9.  125a867c6d2.  19.  28  a  px^y\ 
10.  162i>4j2x7?/9.  20.  3a7m5z4y7. 


SECTION    VIII. 

Art.  22. 

1.  4a26  — a&2. 

2.  3a6c2. 

3.  8  a  b  c3  -J-  a  b3  c  —  6ab3  —  4  m  n. 

4.  — cp3r-{-7m!2ri2-\-7gh. 

5.  2  77*71  r  —  8p3  q  -j-  10  b  c. 

6.  27a;2  — 8y3  +  5m. 

7.  7  «3^?  m  —  3  ap  m3  -{-  a  mp^  -\-  6. 

8.  m  rfip3  r4  +  12  ?rc2  n  —  w2  ti2^3  r4. 


SECTION    IX. 

Art.  25. 

1.  20a  +  26+16c  +  4tf  +  6. 

2.  12a&  +  4cd  +  4ro2n  +  3m. 


X.  KEY.  9 

3    18a&  +  5crf  +  5/w2  +  4cm. 

4.  13a2&2  +  8m2  —  18mrc -f-3&  +  4zy. 

5.  16  6c  — 3a6  +  5ac+12crf  +  4m»  +  10aw. 

6.  8a- S6  +  c  —  17c?  +  25. 

7.  —8a  +  6c—\09d  +  3le  —  Wf+g. 

8.  17  a2 6—  13  a  62  —  10a6  +  18c3d  —  10  c?4—  12m2/*-f 
11  m  w2  +  m  ra. 

9.  8a3  +  14  63  — 7  6x2  — 8cx2  — 22  +  13r— 9Ay2  — 2A-ys 
—  5ky  —  9  hy  -\-7py2. 


SECTION    X. 

Art.  27. 

1st,  2d,  performed. 

3.  Sab  —  2c2  +  36c. 

4.  _6ac-  G«6  +  96c  +  7cc?—  12awi.. 

5.  8m  —  62;—  /—  3d  —  8. 
Art.  28. 

1.  18a2z  —  G&c+IOz2  —  ax  +  b  —  6. 

2.  lOaz3  —  30a2z2  +  49a32;  —  6a4. 

3.  26a6  —  ]86c2  +  2662  +  c2  —  wz2z  +  47. 

4.  7a2zy  — 86z2y  +  30czy2  — 29#5. 

5.  18x2y2  +  48zy  +  462c2  +  crf3  +  c3c?  + 
Art.  29. 

1.  am —  (be  —  dm). 

2.  abc —  a*2  —  (3 my  —  xy). 

3.  a  +  6  —  (c  +  d). 

4.  a2_|_62^_2a5_(a_|_&). 

5    4m2+12m7i  +  97i3  — (2m  +  3n). 

6.  4a6  +  7am-(x2-3f2). 

7.  7m«  +  x2  —  (pq —  y'3). 


io  key.  xi,  xn 


SECTION    XI 

Art.  31. 

1st,  2d,  3d,  performed. 

4.  2<z2  +  5a  — 12. 

5.  a4  +  4M. 

6.  37  a2  b  c*  —  12  a2  e4  +  22  a  c4  d*  —  21  a2  62  _  26  a  6  c2  d» 

—  8c4d4. 

7.  126  a6  c2  +  30  a3  6  c2— 3662c2  +  108<z3c  zy2  —  28a3cz2y 
+  33a3c3  +  72&cz#2+126cx2y  — 246c3— 24  z3  #3  + 2  c2z2y 
+  36  c2  xy*  —  3c4. 

8.  3a6  6_i0a562  +  17a463  —  I7a364  +  8a2&5  —  ab6 

9.  4265  —  21763c2  —  1162c3  +  86c4  —  St*. 

Art.  33. 

1.  a3  +  a»6  — 2a69  — ac»  +  6c8. 

2.  a4  —  4a462  +  8a263  —  464. 

3.  ms  —  n8. 

4.  2a26  +  262c  +  2c3. 

5.  7z3  +  z2y  — 12zy2  — 6?/3. 

6.  10acz-f-2a2c  +  5a&c  +  2z  +  6a  +  8&. 

7.  4a&c  +  2«2rf  +  262d 

8.  4  a  z2  —  4azy  +  ac2  +  462;2  —  4  6  z  y  +  6  c2 —  a3c2w 

—  a  c2  x  y  -f-  a3  J2  m  -(-  a  d2  z  y  —  cPc2mz  —  c2  z2  y  -f-  a2  d2  m  x 
+  d*x2y. 

9.  13  63  c2  z  —  8  62  c3  x2  +  18  c5  z2. 

10.  1 12  b  —  4  6  x  +  a2  6  +  448  —  16  x  +  14  a2  +  0  a2  6  c  — 
7  a2  c2. 


SECTION   XII. 

Art.  40. 

1.  Performed.  4.  12  66  A3 

2.  4  a2  ft  c  J.  5.  25  m5?/6. 

3.  5a263cc/.  6.  31a262?n2. 


XIII 


KEY 


11 


7.  3  m3  a6  bA  y. 

8.  «2/i2w2. 

9.  15. 

10.  9a4m4z3y. 

11.  Slxy*. 

12.  2  a4  62. 

13.  3mz4y3z. 


14.  126  a4  b4  d2. 

15.  37  mnp1. 

16.  23  w2r5s. 

17.  3z2y°. 

18.  7  614. 

19.  79m6ra5. 

20.  103  z. 


SECTION    XIII. 


Art. 
1. 
2. 
3. 
4. 

Art. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 

Art 
1. 

V8' 


5.  3a3&3—  1. 

6.  Sx^y  +  l—Gx. 

7.  _2c  +  l+(3a2&4c» 


49. 

ab-\-m. 
\y  —  2  x  -\-  y~. 
flm2_[_2»i3x  — 3. 
6rf— 46c  —  262. 

44. 

a"2  +  2  a  m  -f-  w2. 

a3  +  3a9x-|-3ax»+s». 

m3_3m2z_|_3mx2__z3# 

2x3— 4x*+8i— Id 

3„2_a6_262. 
4a2  — 5a6  +  262. 

2z2  — 3rtz  +  «2- 

a^  +  atz  +  azz  +  z4. 

a4-|-4a3z+12rt22;2  +  16ax3+16z4. 

x4_x3_|_x2_x_|_l. 

—  5a2  +  4a6  +  362. 

m6  -f-  m5  x  -\-  7ii4  xl  -\-  m?  x3  +  wi2  z4  +  ■■ z5  +  z6- 

5a362c3  — 2a262c4  — 3a62c5  — 7  6c6. 

46. 

i8 + z7  # + *6  #2  +  z5  y3 + z4  #4  + z3  y5  +  x<2  y6  + r  y1 + 


2.  81a4+108a3&  +  144a262  +  192a63  +  256&4. 


12  KEY.  XIV. 

3.  x9  —  x8  y  -\-  x7  y2  —  a;6  y3  -|-  x5  y4  —  x4  y5  -\-  x3  yG  —  z2  y7  -J- 
xy8 —  y9. 

4.  1  —  m  -\-  m2  —  m3  -{-  m4  —  m5-\-m6  —  m7. 

5.  m4  —  m3  n  -j-  m?  n2  —  m  n3  -f-  n4. 

6.  m8  —  m7  -\-m6  —  m5  -j-  m4  —  m3  -\- m2  —  m  -(-  1. 

m12 

7.  1  +  m*  +  m4  +  nfi  +  m8  +  m10  +  -£— -. 

8.  a  —  axy-\-ax2y2  —  ax3y3-\-ax4y4  —  ax5y5-] % 

1-4-zy 
Art.  47. 

1.  a2. 

2.  (*+l)(a  +  6). 

3.  2(x2  +  a;  +  l)  (a  +  m). 

4.  («  +  6)(za  — zy  +  y2). 

6.  3m(i3-^3)  (m2  +  n2). 

7.  5  (m2  —  1)  (x4  —  23  _|_  X2  _  x  _j_  X)# 


SECTION    XIV. 


Art. 

48. 

1. 

abcx 
m 

2. 

b  mx  -\- 

c  mx 

m*  +  2  m  n  +  w2 

5.    ^ . 

6  —  c 

12  a4  +  25  a3  b 


6. 


62  +  2«c 


awi  -|-  bm-j-  an  -\-bn  cfix  —  b2x-\-cfiy —  V*y 

6.    — .  7.     =— j 

c  b-\-c 

Acxy-\-Sx2y  4a?c2  —  4b2c'2 

a  —  x  m2  -f-  2  m  n  -f-  ft2* 

50^a:3  +  26fl3;2y+125«a;4  +  65Q2:3y 

17  6  +  3  c. 

10th  and  11th  performed. 


XIV.  KEY.  13 


Art.  40. 

4a  +  3xy  4  5m2  +  11  qx2  — 93 


---  i  — ^  6 

12xy  —  19x2y2 

n     mt  —  rP  +  ab 

3. ' .  7. 

a  —  b 

14ac  +  2562  +  43  _ 

4.     x : — zl — m ^ .  C5« 


x  —  y 
36  c  -\-4xy 

a~+b      ' 

Sab 
5(x2-*,2)' 
7mx  -j-Sb y 


3a +  2  a2  6  — 6 xy*  2  (a  —  b) 

9  gx  +  6y i 

7  (m5  —  miy-\-m3y'2 —  m2y2-\-myA — y5)' 
15xy 


10. 
11. 


2(a  +  6)(z4  — x3#  +  x23,2  — xy3-j-y4)' 
a-\-b  cm 
3(c  +  d)' 


12  Zac+xy 


(m  -j-n)(x4  —  x3  y  -\-  z2  y2  —  x y3  -|- y4)' 
13th  and  14th  performed. 

15.  a  +  b.  17.    3a26c  +  4xy. 

16.  ro2  +  2mc  +  c2.  18.    46cm  —  12z2y 

Art.  *50. 

a?  cm  (fix^y* 

1.  •  "•      ?c     ^   . 
x  y  »  w 

cPm  „    b&y  —  bcdy 

2.  — j— .  7.  . 

a  x  —  t» 

„     x4y4  0    5a2c  +  4«&c 

3.  — ~.  8. . 

ac2  ?(x  —  y) 

7a*p*q'i  g    a2  +  2ax  +  x2 

x?/3  2(«-|-??i) 

3az-[-36z  a  6c  m2+a2  x-f-62c  ffl3-{-a  6  x 

n  ab  (m~ -\-  t  J 


14  KEY.  XV 


SECTION   XV 


Art,  til. 
1. 


a.  if.  t.  s±» 

116c  x~ry 

3  262y  g  a2  +  fl6  +  62 

4  »t2  —  ra2*  '       36c  —  x  y 

4  17  o  »*»  +  8« 

'    6c  +  4a:y'  '  «6  +  742' 

5.    f±*.  10.       6a;-7 


llm  +  4/i2 
11.  Performed. 


Art.  52. 


1.    —  7.  3  +  7* 


6c»*  4^)  -\-±q-\-bcp-\-bcq 

*    62  — c2'  z3  — y3' 

m  +  n  2m2  +  3a; 
4a?  +  4zay  62  +  c2  * 

3a6  +  46  g«  +  aiy  +  ya 
9fc2  — 16  m2*  «2 

x-f-y  i «     m3  —  m2  n  -f-  w  n*  —  «3 


G. 


m  -\-  ri  x  -|-  y 

*+y 


«  +  6 
Art.  53. 

L    3****  5  a3  63 

2    ?M?  4     -L 

*     7a6*  7atj* 


XVI.  KEY.  15 


7m2rc  8     2(x  —  y)(a  +  6)> 


ra 


6.    ,  0. 


2  (m2  -f  m  n  +  rc2) 


4x-|-4y  5azy 

s  —  y 
21mrc* 


SECTION     XVI. 

Art.  54. 

1.  Performed. 

2.  a(6  +  c  —  2m). 

3.  2x(2a  +  y  +  6). 

4.  5x2(5  +  6y  — 3m). 

5.  33a6c(3  +  a6  +  2a2). 

6.  32m2(llxy  +  12pa  +  2r)- 

7.  22.3a6(l+2c  — 3x). 

Art.  57. 

1.  1,  a,  a2,  6,  62,  a  6,  a2  6,  a62,  a2  62. 

2.  1,  a,  a2,  6,  a 6,  a2  6,  62,  ah*2,  a262,  c,  ac,  cfic,  be,  abc, 
a2  b  c,  62  c,  a  62  c,  a2  62  c,  c2,  a  c2,  a2  c2,  6  c2,  a  b  c2,  a2  b  c2,  62  c2, 
a62c2,  a262c2,  c3,  ac3,  a2c3,  6c3,  a 6c3,  a2 6c3,  62c3,  a62c3, 
a2  62  c3. 

3.  1,  3,  9,  x,  3x,  9x,  r«  3x2,  9x2,  y,  3y,9y,  xy,  3xy,9xy, 
x»  y,  3  x2  y ,  9  x2  y ,  y2,  3  y2,  9  y2,  x  y2,  3  x  y2,  9  x  y2,  x2  y2,  3  x2  y2, 
9  x2  y2,  y3,  3  y3,  9  y3,  x  y3,  3  x  y3,  9  x  y3,  x2  y3,  3  x2  y3,  9  x2  y3. 

4.  1,  2,  5,  10,  25,  50,  m.  2m,  5m,  10m,  25m,  50m,  m2,  2m2, 
5  m2,  10  m2,  25  m2,  50  m2. 

5.  1,2,  3,  6,  5,  10,  15,  30,  25,  50,  75,  150. 

6.  1,  2,  4,  3,  6,  12,  5,  10,  20,  15,  30,  60,  13,  26,  52,  39,  78, 
156,  65,  130,  260,  195,  390,  780. 

7.  1,2,  4,  a,  2a,  4a,  6,  2  6,  4  6,  a6,  2 a 6,  4 a 6,  a  +  4, 
2(a  +  4),4(a  +  4),a(a  +  4),2a(a  +  4),4a(a  +  4),  6(a  +  4) 
26(a  +  4),  46(a  +  4),a6(a  +  4),2a6(a-L4),  4a6(a-|-4). 


16  key.  XVLI,  XV  III. 

8.  1,5,    a,   5a,   2ft  +  5m,   5(26  + 5m),    a(2ft  +  5m), 
5a(2ft  +  5m). 


SECTION    XVII. 

Art.  *59. 

1.  4a2ftc.  6    x_^% 

2-  5a.  •       7.  z3  — 8a;  — 3. 

3-  15  a.  8.  2a  +  2x. 

4-  i»2  +  l.  9.  a  — 6. 
5    x  +  1. 


SECTION    XVIII 


Art.  61. 

1.  Performed. 

2.  36  a3  m4?/3. 

3.  150  a3  m3  a;3. 

4.  max2  if. 

5      (12aft(2ft  +  c),  or 
1  24  a  ft2  +  12  a  b  c. 
27m2x4(l+2n 
2?  m2  a;4  +  54  m3  s* 


(  27  m2  a;4  (1+2  mx),  or 
t  27  m2  a;4  +  54  m3  s* 
i{a  +  bf{a-b),  or 
<  a3  +  o2ft  —  aft2— ft3. 


8      i(«  +  6)(«—  &)(«2+«&+ft2), 
*  a4  +  a3ft  —  aft3  —  ft4. 


XIX  KEY.  17 


Art.  63. 
1.    Performed.  10 


SECTION     XIX. 

26  +  13m2 


144  a 


2.   I*.  11.  ' 


5y'  "    3fli-2a2' 

3a3c  3am2 

t>.      "^ ^.  X/i, 


2m2'  *  a2  —  2m2 

4.  ^^0.  13.  Performed. 
15xy2 

5.  =     *.  14.      *y 


12  mxy'2  x~\~y 

6.  4-.  15.   a-±\. 
t  a6m  a  —  o 

„    aro  +  35x  %a    5(a-f6)        5a  +  56 

7.    ^ .  16.       VQ       ',  or  — ^ 

8.  *y+2.  17#  «2+«*+*2 


3  x2  y2  '  a 


x 


9      25  an  c^  3(x-y) 


5a6c  '    4(z2  — iy  +  y'2) 

19.  9"^ 

5  (x3 —  x2y-|-xy2  —  y3)' 

iRT    64. 

1.    Performed.  5. 


+  n 
2.   J2-.  ft  »■ 


2a6c*  *    2xy  +  3y2' 

3.  JL.  7.         i 


2xy'  '    2(x  — y)' 


c 
4~^2' 


8.    * 

1 


9. 


2* 


3(x2  +  xy  +  y2)' 


IS 


XX,  XXI, 


Art.  ©*5. 
4ab4 


1. 


2. 


SECTION    XX. 


9x 


21c 

48  S3?/3 

35  6  c  m2* 


12  s3  +  9  &2  s2 
76  +  9m2     ' 
J     8z2  +  2z 
4'    7—  * 


5. 


6. 


3a3  —  3a26 

56 
•    3s  —  4 
4  a  x2  —  6  a 


5  c  x2  y 


9. 
10. 
11. 
12. 


y 

Slab  ex  — 21b  ex* 


a 

• 

4 

ah  c  x  -f-  46  c 

z2 

3 

a?m?  —  3a?n2 

2; 

3 

a2  m  x3  +  3  a2 

#z  y3 

SECTION    XXI, 


Art.  68. 


A  w  a 
_     29  a  y  +  35  x 


24a62-f953s  +  22  a8  y 
42  a2  63 


35y 

62  +  9&3 
42  a2 
27  q*  x2  y  +  6  a  b  p*  +  2  p  x  y* 

12  p3  g2 
5  a  x  yz  -|-  5  b  x  y~  -f-  33  s2  y  -f  117  a 

225  x3  w2 
17  a  —  4  5 
162       ' 


XXI.  KEY. 

63  m  re4  x  y  +  70  b  c  n3  +  60  h  x 

525  77*3  w4 

150  m  x*  y2  +  70  b  c  x  \f  +  35  c2  y2  +  56  a  x» 
8t  350x3^2 

105  a3 +  364  a2 +  390  6 
9'  910  a 

io.  *££ 

1 X4 

i.i.      2 

12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 


1  — x2 

3  +  7a  +  76 

a*  —  b*       ' 
3a26  +  6a2  +  562»i 

4a&3  +  8a&2 
14  6  —  9  am 


12  m2 
3a2-}- life2 

2 

196 

a  ex 

/j2  —  C2* 

4x2  +  3 

17 

3x     * 

«2_26 

252 

32  x  —  3  ax  +  7a 

56  a2 
4a2x  — Sax2  — 6as 
21a  — 42x 

4x2 


1  — X4 

b  c  +  mjpx  +  m  j>y 


2C  key.  XXII. 


23 

5a+36 
4       * 

9A 

11  m3  4   7  m  n2  —  2  m2  w  +  ra3 

7m  —  2w 

25. 

4«2  —  2  62 

a2  +  62 

„fi      12  m2  x  y  —  3  a  -\-  7  b 

4Lx  y 
27     5  h3  n3  x  -\~  h3  n3 


28. 


x+l 
7«6  +  4c2  —  2m3 


m 


29      9x~\-4bc  —  a2  —  ab  —  ax  —  bx 


a-\-x 

x  -\-n 
x  —  y 


30.    *2  +  y2  —  mx+my  +  x  —  y 


SECTION    XXII, 


Art. 

©9. 

3m 

1. 

~sT 

2. 

7ac3 
3     ' 

3. 

4rc(x2  +  #2) 

m2 

4. 

5c(a  +  5) 
2 

5. 

5m 

6. 

7. 
8. 
1). 


a3  a:5 
Wh*73' 

May 

bmx' 
4bx 
81a3mp' 
W(x  +  y) 
3 


» * 


XXIII.  KEY.  2l 

9msV(l  +  36) 
1D'         4(26  c  +  6)      * 
31a(22m+j?) 
429  6  jp2 

18  l"bc 


1 1 

4(*-3). 

11 

5       * 

1° 

9z  — 3 

X 

13. 

4m 
ab' 

14. 

1  my ' 

i* 

2m 

19. 


20. 


11  %2{m — p) 

ox(m—l)' 
56c2 


3x2(z-  y)'  4(z+y) 


SECTION   XXIII 


Art.  70. 

1.    Performed. 
a2  +  62c 


2. 


a-\-bc 


Sbcd —  ad-\-  be 

6  +  6  dm 
bem  +  c^m  —  362  +  66c 
8c  — 46  +  6c?  +  cd     ' 

2a6  +  36c 


5wi2  +  3a  +  12c* 
a  c2  —  ab  c-\-bcd —  ftd 
a?  —  b*  +  bc  ' 

c' 

762c+  150  ^  —  336  +  6  abd 
Sad-\-7bc  —  33 

616  —  30  a 

186  —  9fl' 


22  key.  XXIV,  XXV 


SECTION  XXIV. 

Art.  74. 

1.  1st  8;  2d  3. 

2.  1st  $12;  2d  $16. 

3.  Cow  $25  •  ox  $35. 

4.  A  $12;  B  $20. 

5.  Potatoes  2s.;  corn  $1  per  bushel. 

6.  Wheat  10  bushels ;  rye  8  bushels.    Sold  5  and  3. 

7.  Father's  $1 ;  son's  2s. 

8.  f 

9.  TV 

10.  He  bought  200  lbs.  of  coffee  and  300  lbs.  of  tea;  he  sold 
160  lbs.  of  coffee  and  200  lbs.  of  tea. 

11.  200  eggs  at  2  for  a  cent ;  100  at  5  for  3  cents 

12.  Smaller  75,  larger  100  gallons. 

13.  Wheat  9s.,  oats  3s.  per  bushel. 

14.  Larger  160,  smaller  80  gallons. 

15.  Upper  part  15  feet ;  lower  part  25  feet. 

16.  A  $500;  B  $700. 


SECTION    XXV. 

Art.  75. 

1st,  2d  and  3d  performed. 

4.  Flour  $7,  rice  $8  per  barrel;  sugar  $10  per  box. 

5.  1st  2  ;  2d  3  ;  3d  5. 

6.  A  $2;  B$0-75;  C  $050. 

7.  1st  $500;  2d  $600;  3d  $800. 

8.  A's  60,  B's  80,  C's  90  years. 

9.  1st  5s. ;  2d  6s. ;  3d  7s. ;  4th  8s. 

10.  A  $0-30,  B  $040,  C  $050,  D  $060  per  lb. 
11    Wheat  $2,  rye  $1,  barley  $1£,  oats  $£,  potatoes  $±  per 
bushel. 


XXVI,  XXVII.  key.  23 

SECTION   XXVI. 
Art.  76. 


1st  and  2d  performed. 

18. 

425. 

34. 

1639. 

3. 

2100. 

19. 

90. 

35. 

561. 

4. 

10500. 

20. 

48. 

36. 

561. 

5. 

12600. 

21. 

880. 

37. 

121. 

6. 

4500. 

22. 

1011 

38. 

8470. 

7. 

35000. 

23. 

4. 

39. 

4. 

8. 

f 

24. 

—6. 

40. 

2. 

9. 

21. 

25. 

112. 

41. 

24. 

10. 

140. 

26. 

180. 

42. 

o. ; 

•  11. 

». 

27. 

115. 

43. 

18. 

12. 

A- 

28. 

510. 

44. 

1. 

13. 

18. 

29. 

187. 

45. 

1. 

14. 

37. 

30 

33. 

46. 

2. 

15. 

47. 

31. 

—  33. 

47. 

f- 

16. 

100. 

32. 

3. 

48. 

—  1. 

17. 

220. 

33. 

187. 

49. 

h 

The 

riven. 

)  remaining  answers  in  this  article 

being  indefinite,  are  not 

SECTION    XXVII, 


4rt.  77. 

1st  and  2d  performed 

11th,  12th  and  13th  performed. 

3.  60  and  90. 

14.  9  oxen,  45  cows,  315  sheep. 

4.  90  and  140. 

15.  Performed. 

5.  540  and  660. 

16.  2-fT  days. 

6.  11  and  16. 

17.  3-^y  hours. 

7.   16  and  19. 

18.  Performed. 

8.  33£  and  36£. 

19.  A  $200;  B$150;  C  £100 

9.  17£  and  30f 

W.  A  $54;  B  $27;  C  $18. 

10.  32|  and  Gfy 

21.  $200;  $150;  $100;  $50. 

24.  KEY.  XXVII. 

22.  A  $128;  B  $80;  C  $64 

23.  Performed. 

24.  $691875. 

tr 

26.  $75. 

27.  r=— . 

tp 

28.  -055  or  5£  per  cent. 

29.  t  =  — . 

30.  7  years,  1  month. 

31.  Performed. 

32.  $8620625. 

33.  p  =  — — . 

34.  $250. 

tp 

36.  *065  or  6£  per  cent. 

37.  t  =  a-=P. 

rp 

38.  3£  years. 
39"   TXT' 

\-\-tr 

40.  ,£1  4s.  8Jd. 

1  2 

41.  £  =  - ,  doubled ;  t  =  -,  tripled, 

r  r 

42.  16f  years,  doubled  ;  33J  years,  tripled. 

43.  20  years,  doubled ;  40  years,  tripled. 

.  .  a  an 

44. 


45.    1st  — — ;   2d 


46. 


-\-n  m  -\-  n 

a  am 


\A-m-\-n'    \-\-m-\-ni    \-\-m-\-n 


XXVII.  KEY.  25 

bed — ab      ac —  bed 


47. 

48. 
49. 


c  — 6      '        c  —  b    ' 
am-\-bmn      an  —  bmn 

m  -f-  n      '        m-\-n 
mnp 
n  —  rri 


amn 

50.    guineas. 

mn  —  m  —  n  ° 

51.     ^ dollars. 

nq  —  mq  —  np 

~~       ab     , 

52.  -, days. 

6 — a 

53.  ±L. 


5a        g(4  +  6)     g(3  +  26)     a (2  +  36)    a{l+4b) 
54-    6_i;      6—1    ;       6  —  1      ;       6  —  1     ;      6—1     J 

5ab      r 


6  —  1' 

55.  ——  shillings. 

~n     t,t         ,     c-4-bn  .  ...an  —  c  . 

56.  Wrought  — ~ — =-  days  :  idle ; — r-  days. 

to       a  +  6        J    '  a-\-b       J 

57.  — ! —  beggars. 

,Q       A,     «  +  &  B,    « — b 

58     As  — - —  years ;  Bs  years. 

59.    Boots dollars;  shoes dollars. 

ac  —  1  ac  —  1 

be  —  1  be  —  1 

ivt  ap(m-\-n)      .  .  an(p-A-q) 

61.    Numerator     *v      './ ;  denominator  l£_i_l/. 

mq  —  np  mq  —  np 

__     __  ap(m-\-n)  .  an(p-\-q) 

t)2.    Numerator  -*-* ! — L\  denominator  vr  '    -. 

np —  my  rap —  mq 

3 


26  KEY  XXVIII. 

63.  Particular  answer  to  61st  f ;  particular  answer  to  62d  |£. 


SECTION    XXVIII. 

Art.  84. 

From  the  1st  to  the  5th  inclusive,  performed. 

6.  This  question  as  enunciated,  if  x  represents  the  number, 

gives 

7x       5i    .    . 

From  this  equation  we  find 

x  —  _350. 

Changing  the  sign  of  x  in  the  original  equation,  we  have 

7x            5x   .    m 
— :  —  — —  -\-  5  ;  or,  by  transposition, 

To  rectify  the  question,  therefore,  we  enunciate  it  thus  • 
What  number  is  that  %  of  which  exceeds  y7^  of  it  by  5  ? 

7.  This  question  gives  the  number  of  years  =.  —  40;  and  the 
equation  being  modified  accordingly,  we  find  that  the  question 
should  have  been : 

How  many  years  after  marriage  was  his  age  to  hers  as  7  to  6* 

x 

8.  If  -  represent  the  fraction, Hhe  question  gives 

y 

From  these  equations  we  find 

as  =  —  3,  and  y  =  — 13. 

Changing  the  original  equations,  in  a  manner  similar  to  that 

pursued  in  the  solution  of  the  5th  question,  we  have 

%  x  4-2 

=  i;  and  — - —  =.  -j-5^;  from  which  the  modifica* 


y  +  2-5  y 

♦ion  to  be  made  in  the  enunciation  is  manifest. 


XXIX.  KEY.  27 

9.  Let  x  =  the  greater,  and  y  =  the  less ;  then, 

x  —  y  =  20,  and  6x  —  3y  =  96. 
These  equations  give  2;  —  12,  and  y  =  —  8. 
Changing  the  sign  of  y  in  the  original  equations,  we  have 

x  -\-  y  —  20,  and  6  x  +  3  y  =  96 ;  and  the  ques- 
tion should  be  changed  accordingly. 

10.  Let  x  and  y  represent  the  number  of  gallons,  which  flow 
through  the  cocks  A  and  B,  respectively,  in  a  minute.     Then, 

5x  +  3y  =  24,  and7x  +  5y:=32. 
These  equations  give  x  =  6  gallons,  and  y  =z  —  2  gallons. 
Changing  the  sign  of  y  in  the  original  equations,  we  have 

5x  —  3y  =  24,  and  7x  —  5y  =32. 
Hence  it  appears  that  water  flows  out,  instead  of  flowing  in, 
through  the  cock  B. 

11.  A's$5000;  B's$3000;  C's  —  $2000. 

Hence,  either  C  is  in  debt  $2000  ;  or  he  possesses  $2000,  and 
a  certain  number  of  times  his  estate  is  subtracted,  in  each  case, 
instead  of  being  added. 


SECTION    XXIX. 

Art.  85. 

1.  Let  x  represent  his  money  in  cents.     Then 

^  +  40  =  ^  +  49. 

This  equation  gives 

3x  —  3x  =  9;  that  is,  (3  —  3)x  or  0.x  =  9;  therefore, 

T  9 

X jj. 

Hence,  the  question  is  impossible,  or  the  value  of  x  is  infinite. 

2.  Let  x  =  A's  age ;  then,  x  +  10  =  B's,  x  +  20  =  C's,  and 

1  —  34  =  D's.     We  have,  therefore, 

s  +  10       2z  +  40       5x-170 

2+3       -\  e         -ix-W. 


28  KEY.  XXX. 

This  equation  gives 

12s—  12z  or  0.z:=60  —  60;  and 
*  =  &. 

Hence,  the  value  of  x  is  indeterminate;  consequently,  their 
ages  cannot  be  ascertained  from  the  conditions. 

3.  If  x  represent  the  number  killed  to-day,  and  y  the  number 
killed  yesterday,  we  have 

x       y    ;     •        .  3a; 

These  equations  become,  by  multiplication  and  transposition, 

3  x  —  2  y  =  30,  and  3  x  —  2  y  =  10,  in  which  the  first 

members  are  identical,  while  the  second  members  are  different. 

Hence,  the  two  conditions  are  incompatible  with  each  other. 

By  subtracting  and  canceling  the  terms  containing  y,  we  have 

3  x  —  3  x  =  20,  or  0  .  x  =  20,  and  x  =  %<*. 
Or,  by  subtracting  and  canceling  the  terms  containing  x,  we 
have 

2y  —  2y  =  20,  or  0  .  y  =  20,  andy  =  %0-. 
The  problem,  therefore,  is  impossible;  or  the  values  of  x  and 
y  are  infinite. 

Other  forms  for  the  answers  may  be  obtained  by  different 
methods  of  elimination ;  but  they  are  all  essentially  the  same, 
because  they  are  all  infinite. 


SECTION    XXX 


Art.  86 

1.  28. 

2.  53. 

3.  78. 

4.  29. 
Art.  87. 

1.  37. 

2.  49 


5. 

36. 

6. 

93. 

7th  and  8th 

performed. 

3. 

19. 

4. 

351. 

X3 

:i,  xxxn. 

KEY. 

5. 

709. 

9. 

1197. 

& 

109. 

10. 

12325. 

7. 

190. 

11. 

30021. 

8. 

1049 

12. 

50309. 

29 


SECTION   XXXI 


Art. 

89. 

1. 

t- 

2. 

h 

3. 

* 

A.RT. 

90. 

1. 

1- 

2. 

*-> 

3. 

A-- 

Art. 

93. 

1. 

1-4142 +. 

2. 

51962—. 

3. 

5-8095—. 

4. 

12 1370 +. 

5. 

58676—. 

6. 

180537 +. 

4. 

tf 

5. 

M- 

6. 

Mr 

4. 

A+. 

5. 

B+. 

6. 

e-. 

7. 

0-8528 +. 

8. 

0-8292  — . 

9. 

0-8660  +. 

10. 

003937  +. 

11. 

0-039 15+. 

12. 

201008 +• 

ECTION    XXXII. 

Art.  93. 

1.  A's  21,  B's  27  years. 

2.  A  10,  B  12  miles  per  hour.     A  went  1  )0,  B  120  miles. 

3.  5  yards  and  6  yards. 

4.  10  rows  ;  12  trees  in  a  row ;  960  bushels. 

5.  30  miles. 

6.  50-662  -f  inches. 

3* 


30  KEY.  XXXIII 

7.  12-649  +  rods. 

8.  43-818—  rods. 

9.  16. 

10.  Greater  60;  less  15. 

11.  Length  30,  breadth  25,  height  20  feet. 

12.  37  yards  in  each ;  prices  42s.  and  32s.  per  yard. 

13.  30  persons;  $150  each. 

14.  Greater  57 ;  less  45. 

15.  $150. 


SECTION     XXXIII 

Art.  95. 

1.  By  the  2d  statement,  9  yards. 
Art.  97. 

1st  and  2d  performed. 

3.  15  rods. 

4.  9  rods  ;  $3  per  rod. 

5.  Greater  90  ;  less  85. 

6.  Length  100,  breadth  60  rods. 

7.  10  sheep ;  13  calves. 

8.  40  hours ;  4  miles  per  hour. 

9.  $100. 

10.  80  feet. 

11.  Coffee  $0-10,  tea  $0*50  per  pound. 

12.  4  rods. 

13.  225. 

14.  5  sons. 

15.  12  fowls. 

16.  A's  stock  $100. 

17.  2  rods. 

8.  10  gallons. 


XXXIV,  XXXV,  XXXVI.  key. 

SECTION    XXXIV 

Art.  lOO. 

1.  Performed.  6.     335. 

2.  67.  7.     364. 

3.  39.  8.     909. 
4     9L  9.     3009 

5.     77. 


SECTION   XXXV. 


Art.  101. 

1.  i- 

2.  TV 

4. 
5. 

23 

27* 

ag.  or  4}. 

3.     TV 

Art.  102. 

i.    H+- 

3. 

AV+- 

2.    iM— 

Art.  103. 

1.  1-2599+ or  1260— . 

2.  19129+  or  1913— . 

9. 
10. 

0-956 +. 
0-7688+  or  0769 

3.     4-8629+  or  4863—. 

11. 

1-957 +. 

4.     11447+  or  1145—. 

12. 

2-3339  +  or  2334 

5.  2951  +. 

6.  0-292  +• 

7.  2-3129+ or  2-313—. 

8.  10-790+. 

13. 
14. 
15. 
16. 

1542+. 
0183 +. 
2044 +• 

2-823+. 

SECTION 

XXXVI 

Art.  104. 

1.  4,  6  and  10. 

2.  Length  9,  breadth  5,  depth  7  feet. 

32  key.  XXXVII,  XXXVIII 

3.  5  and  10. 

4.  Length  30,  depth  8,  breadth  22  feet. 

5.  16  feet. 

6.  18  feet. 

7.  6  and  12. 

8.  22-071+ inches. 

9.  Side  of  the  base  31*072+  feet;  altitude  124288+  feet. 

10.  Radius  2  feet;  length  7  feet. 

11.  1-8837+ feet. 


SECTION    XXXVII, 


1. 

49  a2  m4. 

2. 

64  64c2z8. 

3. 

225  a8  m6pu. 

4. 

128z143/21. 

5. 

&26c78^91. 

6. 

1024  630  C20J90. 

7. 

Performed. 

8. 

pmnqmt. 

9. 

Performed. 

10. 

ZmpmXqmy. 

11. 

81  »V. 

12.  —32x^35. 

13.  —  343aH6c3^. 

14.  64  a6  m18  rc12 jp30  ?42  x6  y* 

15.  Performed. 

16.  ^ 
17. 
18. 


125m6rc3 
216  &lp 
1 


81p8x 


8- 


SECTION     XXXVIII 

Art. 

106. 

1. 

2. 
3. 
4. 
5. 

(6m — n-\-pY 

(b  +  c  +  d)™. 

(4a6  +  4zy)14. 

(x-yf* 

(a  +  6)2  (a  — 6)6 

XXXIX.  key.  33 

6.  243  (x  —  y)15(p—  qf5. 

7.  S(a  +  b  +  c)^. 

8.  a*m*\c  —  d)*m(z  +  yy\ 

9.  (a  +  6+c)3OT. 

10.  (a+bfn  {c  —  dfn. 

11.  (z  +  2#r. 

12.  (a  +  x)m?  (n  —  x)mq. 
13th  and  14th  performed. 

(b  +  c)* 


15. 
16. 
17. 
18. 


(ra  +  rc)6 


(i>  +  ?)9* 

l6(ft  +  &)4(s  — y)8 

81  (C  +  d)32  ' 

a7(m^_w)Ql(a:_y)35 

V(r  +  Sy*(a  +  b  +  c)21' 

Art.  107. 

1.  »i2  —  2»ix  +  x2. 

2.  a3c  +  3a2&c  +  3a62c  +  63c. 

3.  o2»z-|~2«^»i  +  ^2^  +  2acm-|-26cm-|-c2/M+aaw   f- 
2a6»+62w  +  2acra+26cn  +  c2w. 

4.  a3  x2  -\-  2  a3  a:  y  -f-  a3  ?/2. 

5.  4c2+12cfl?  +  9d2. 


G. 


c3_j_3c2^^.3c^_|.rf3- 


SECTION    XXXIX. 

Art.  113. 

1.  a7  +  7a66  +  21a562  +  35a463  +  35a364  +  21«265^ 
7  a  66  +  b\ 

2.  x10  +  10x9y  +  45x8y2  +  120x7y3  +  210xV  +  252x5y 
+  210  x4  3/6  _|_  120  x3  if  +  45  x2  y8  +  1 0  x  y»  +  y ™. 

3.  m5  —  5  ra4  w  +  10  m3  w2  —  10  m2  n3  +  5  m  n*  —  n5. 


34  key.  XXXIX. 

4.  611  +  11610c  +  55&9c2  +  165&8c3  +  330  67c4  +  462#c5 
+  462  b5  c6  +  330  M  c7  +  165  b3  c8  +  55  62  c9  +  1 1  6  c10  +  c11. 

5.  a;l8  —  13  x12y  +  78  x11  y*  —  286  x10  y3  +  715  x9  y4  — 

1287  z8y5_|_  1716  z7yi_  me  26y7_j_1287z5y8  —  715x4y9  + 
286  x3  y1  °  —  78  x2  y^  +  13  a;  y12  —  y13. 

6.  Performed. 

7.  x5  +  10x4y  +  40x3y2  +  80x2y3  +  80xy4  +  32y5. 

8.  216  a3  +  540  a2  x  +  450  a  x2  +  125  x3. 

9.  Performed. 

1 0.  32  a5  +  240  a4  x  +  720  a3  x2  +  1080  a2  x3  +  810  a  x4  + 
243x5. 

11.  64  x3—  144x2y  +  108x3/2  —  27y3  +  48«x2  —  72axy 
+  27ay2+ 12  a2  x  —  9  a2  y  +  a3. 

12.  a3  +  3  a2  b  +  3  a  62  +  b3  +  3  a2  c  +  6  a  6  c  +  3  52  c  + 
3  a2  d  +  6a&d  +  3  62d+3a  c2+6acd  +  3ad?  +  3b  c2  + 
66c^+36fZ2  +  c3  +  3c2^  +  3crf2  +  ^. 

13.  a6  +  12  a5  6  —  6  a5  c  +  60  a4  62  —  60  a4  b  c  +  15  a4  c2 
+  160  a3  b3  —  240  a3  62  c  +  120  a3  6  c2  —  20  a3  c3  +  240  a2  b4 

—  480  a2  63  c  +  360  a2  62  c2  —  120  a2  6  c3  +  15  a2  c4  +  192  a  b5 

—  480  a  64  c  +  480  a  63  c2  —  240  a  62  c3  +  60  a  6  c4  —  6  a  c5  + 
64  &6  __  192  65  c  _|_  240  b4  c2  —  160  b3  c3  +  60  62  c4  —  12  6  c5 
+  Ce. 

14.  a$ +  5  a4b +  10  a3^ +  10  a?b3  + 5  ab4  +  b$  — 10  a4  c 
_40a36c_60a262c— 40a63c— 15a4d—  60a36d  —  90a262d 

—  60«63rf  — 15  64d  +  40a3c2  +  120  a2  6  c2  +  120«62c2  + 
40b3  c* +  120  a3  cd  +  360a*  bed +  360  ab*cd  +  120b3cd  + 
90  a3  d?  +  270  a2  6  d2  +  270  a  62  d2  +  90  b3  d>  —  80  a2  c3  — 
360a2c2rf  —  540  a2cd2  —  270  a2  d3  — 160a6c3  — 720a6c2d 
— 1080  a6  erf2— 540  a  6  d3  —  80  62c3  —  360  62c2rf— 540  62c^ 

—  270&2d3  +  80ac4  +  480ae3rf+1080ac2d2  +  1080acd3 
+  405  a  d4  +  80  b  c4  + 480  bc3d  +  1080  b  c2  <P  +  1080  b  c  d3 
+  405  b  d4  —  32  c5  —  240  c4  rf  —  720  c3  d2  —  1080  c2  rf3  — 
8l0crf4  — 243^5. 

Erratum.     In  2d  line  of  14th,  insert  —  10  b4  c. 


XL,  XLI,  XLII.  key 

SECTION     XL 


Art.  115. 

1.    5. 


35 


4. 

73. 

5. 

211 

6. 

13. 

2.  12. 

3.  73. 


SECTION    XLI. 

Art.  117. 

1.  ±a?m*z.  7    3s3y 

2.  ±8*y.  •    '     4m4* 

3.  7a3p</2.  8         2q2fe3 

4.  _9a26c4.  5z#2 

5.  i2a264c3.  3a263c 

2a  '     4x4^5' 

6'    *** 

Art.  119. 

In  this  article  the  answers  are  indefinite,  and,  therefore,  the, 

are  not  given. 


SECTION    XLII. 


Art.  130. 

1.    3x2  +  2x  +  5. 

5. 

2x2  +  2ax  +  462. 

2.    a2_|_6«6  +  962. 

6. 

X3-f-2x2  — Z  +  2. 

3.    3x3  — 2x2  +  x— 1 

7. 

2x2  +  ^  +  5. 

4     3a2  —  2a&  +  563. 

^  2     ■ 

Art.  131. 

1.    3a  +  3x. 

4. 

2a2— 63. 

2.    2x3  +  5a2. 

5. 

3x  +  2y: 

3.    5c2—  2yz. 

36  key  XLII1,  XLI* 

SECTION    XLII1. 


i2.  ifd(obd)i 


RT, 

122. 

1. 

SaWs/Wab. 

2. 

4  c2  (5  a  6)* 

3. 

4. 
5. 

3  a3  62  y/Td*. 

3bc\/5^ 

Performed. 

3 

6.    2a^/3«  —  b. 


3. 


13.  £c^/2a262. 

14.  £b(6abrf. 


a 


15.    f  t/3£c. 
4cK 


7.    a(262  +  a26c)*  «>.    g-cy/    -3; 


2  62 


8.    «^/l  +  63.  17      7a%    /ZScfib—lab 


17   Ifl  /^L^ 

5  my       20m- 


9.    Aafotf—\.  bm\       20in»— 10w» 

10.  24i/6«36_3.  ,0     i^«    y/5b+lQa* 

8      '    3  c  K        c  w  +  2  m2* 

11.  -^afc 

Art.  123. 

1.    y/9a26z2.  5.    (12  a2  6)2. 

2     ,     /20  6  c  6     (125 1  y)* 

•     — ? — -  7.    (le^ft^  +  aa^^i 

4     X  8(a— 6)  /27a3f«2w 


SECTION    XL1V 


Art.  125. 

1.  4  a2  (3  z)*.  3.   5(5)*. 

2.  14(2)*.  4    23(3)*. 


XIJV.  KEY.  37 


5. 

8  (2)* 

6. 

8(2)*. 

7. 

5a$  +  &ab* 

8. 

ii(l#. 

9. 

fa  (10)* 

10. 

£(l3  +  2c)(ab)}  +  (xy)i. 

11. 

(2)* 

12. 

2x(3a)*. 

13. 

4  a  (2  6)*. 

14. 

2a(3a&2)* 

15. 

»(«)*. 

16. 

(5)*. 

17. 

(aj)i_|_(10c  — 7)  (m)* 

18. 

Ty(3  a  &)*  +  ,&  (25  m)* 

19. 

21  a  62. 

20. 

10a*  6*  A  or  10a  6  c  (a  6   \* 

21. 

3  ro2  a"  c** 

22. 

75  x^  y* ,  or  75  y  (x5  y  )* 

23. 

Performed. 

24. 

30  (10)* 

25. 

Performed. 

26. 

8  (216)* 

27. 

140. 

28. 

m-\-n 

a  »• 

29. 

mq  +  np 

15  a    »?    . 

30. 

(648000)^. 

38  key  XLIV 

81.  (a  +  b)&. 

32.  12a(c  —  d)^ 

33.  12  a2  6  (« +  &)*(«— yjM 

34.  35(m  +  «)£(c  —  d)&. 

35.  4. 

36.  3—17(6)* 

37.  41. 

38.  a*. 

39.  J  6*  A 

40.  2a&b$c. 

41.  !«*&*<£ 

42.  6. 

43.  15. 

44.  8(2)*. 

n — m 

45.  a  mn  . 

46.  Ja*p    6** 

47.  «  (2,* 

48.  («  —  &)*. 

49.  lto>*! 

a 

50.  («  +  »)*(«-««)* 

3m 

51.  4a%bK 

52.  125  a^c. 

53  8^ 


27x2y3 
54.  A  (3)*. 


XLIV.  key.  38 

288  £  b5  c10 


55. 


m 


56.  (a +  6)*. 

57.  27  (z  —  y)i 

59.   32  (**  —  **)*(*  —  «)*. 


59.  a*  b"^  cm. 

60.  a^  P"  c*  <P\ 


61.    8(fl  +  &)1 


27  (c_^ 

62      4(x  +  y)*(C-rf)*^ 
25(m-f  7i)£(z— y)* 

63.  a*  6* 

64.  3c£b*. 

65.  2*  a*  6* 

66.  3*a*6*c*. 

i    J      I    JL 

67.  10m a*mzm  y®* 

JL    JL    JL 

68.  6m  a«m  bsm 

69.  (a +  6)* 

70.  4  (a  —  6)*(c  —  d)*. 

71.  2a(m-n)^(c-^. 

72.  3*  (« -*)*(*  +  «*)* 


40  KEY.  XLV. 


SECTION    XLV 

Art.  132. 

1. 

7\/% 

2. 

Q\/b: 

3. 

(6«  +  5)v/S 

4. 

8^47 

5. 

43^/37 

6. 

wr°- 

7. 

191  yU 

8. 

¥&% 

9. 

i(3«  +  4^)v/I 

10. 

-13^3. 

11. 

8^2. 

12. 

8|/T— y^a 

13. 

«&(3a  +  5)^/2^6. 

14. 

(a  —  c)  i/5  c. 

15 

/a(fl</c  +  6ce-6^2)\   /F- 

X(J. 

V                b*de                /**'' 

16. 

sy% 

17. 

4v/tT 

18. 

2yU 

19. 

va 

20. 

2^5. 

21. 

Ay^ 

22.   y/2. 


XLV- 


kit. 


41 


23. 
24. 
25. 

26. 

27. 

28. 

29. 
30. 
31. 
32. 

33. 

34. 

35. 

36. 

37. 
38. 
39. 

40. 

41. 

42. 
43. 

44. 
45. 

46. 

47. 

48. 
49. 


2  a  (2  a  —  z)\/5x. 
2a  (4  —  a)  fa 
*  (fa -[/*)• 


tW75' 


8y/2. 

£V/M-rVv/6 

12. 

4. 

2. 

6a^/6c. 

6  — 

5  a2  c  y'a  c. 

30^/10. 

6^432. 

4a2  — 96. 

21  + 18/3"  +  28  fa  +  24  ^"108. 

l-V/5. 

6  c  ^/a13. 

12. 


72y/23328 

24  ^rpr^. 


4  2 


iV/«)2l(7)1466. 

2. 
4* 


*2 


Ki  jr. 


XIV 


50.  4\/b. 

52.  12^/27 

53.  f^/T0. 

54.  ^ 


55.    |^/a262. 


56. 


si«v/4*v* 


57. 

a2  +  2«y/&  +6 

a2  — 6 

58. 

1(10-^/2). 

59. 

KV/3-1). 

60. 

^ 

61. 

26^5? 

62. 

9^/a6. 

63 

64^/aar. 

64. 

a4  5  c  ^/6T 

65. 

«3  63  mio  ^££ 

66. 

xm  ym  i/am  bm 

67. 

\/xy- 

68. 

»vA 

69. 

*v«2. 

70. 

3. 

71. 

V/(«  +  *)10   or  (a-f  ** 

72. 

2^^ 

73. 

18 

3^/a6 

XLVI.  key.  43 

74.   4^/^  78.   \/(a  +  b)K 


75.  2y/a6c2.  79.   \/{a  —  6)9. 

76.  2/^6.  80.   \/x  +  y. 

77.  (/W  81.   V?9(a  +  6) 


SECTION    XLVI 


Art.  134. 

1.  7m7n-5x2. 

2.  12«-6  6-5c5. 

3.  30a*&M. 

4.  100  a*63A 

5.  1. 

„  9  a2  c  w3 

6-  "I— 

7.  14a37»_1  r"1xy. 

^  49  a-3  b~~ J  c~3  m~3  z2  ?/ 

8'  5 "• 

4  (C- «*)(«  + 6)- 

J.  g  . 

49a36(x  +  3/)2(m  — n)(m-f  w)-1^  — <f)"^ 

10.  5 

(T2m-1 


IX. 

2      " 

12. 

5a32r9y"2 

3 

13. 

a56-1  c~~1mz1 

4 

14. 

aa3^3^^-1^. 

15 

7a-1&-2c-22f« 

44  kev.  XLVII,  XLV1II 

16    ZHa  +  b)(x  +  y)(c-J)r^ 

m 

22  ar*  (2  c  — 3rf)-3  (m  +  n)-i  (x— y)-i 
i/.  -  . 

18.  9a-26-*c6. 

19.  64«~666c-9, 

20.  16a26-*c-*. 

21.  1000  m-£  x-1  y\ 

22.  4a-i&-2C3. 

23.  2«-36c"2. 

24.  3a*b-±  c-*d~*. 

25.  3  a*  ft-**-* 

26.  S^a-^a^y-*™-1*5. 


SECTION    XLVII, 


Art.  136. 

1.  Performed 

2.  20. 

3.  51. 


SECTION  XLVIII.  ' 

Art.  137. 

1.  16.  4.  12  J. 

2.  5jh  5.  6. 

3.  5  "and  15.  6.  1st  term  10 :  3d  18 


L,  LI  KEY. 


45 


SECTION    L. 

Art.  143. 
1.  Performed. 

3.  n  = g ,  S_  22 

2S-m      ,      2(S—  ara) 

3.     / J     I* /  ,  \ 

w  w(n  —  1) 

4-  w-a  +  r  a-  2£—  (J+a)' 
5.  Performed. 

_^_t=v/(2^~rf)2+8^      _  d— 2a=fc\/(2a— d)«+8f/ig 
6-  l-  2  ;W""  2d 

2S—dn(n—l)     .     2£  +  c/n  (w  — 1) 

7.  a  = = ;  *  — o~ . 

2w  ^« 

v       ■      2rcZ—  dn(n  —  1) 

8.  a=Z—  d(7i  —  1);  S=z jp ^. 

2S—nl     '       2(n/— fif) 

9.  or j  d=— -r-. 

n  n(n  —  1) 


_<?=Fy/(rf+2Z)2  —  8dS      __  <7-|_27dLy/(^+2/)2— 8</£ 
10.  «_  ^  ;w~  2d 

Let  the  learner  verify  the  preceding  formulae,  by  making  a  =  1, 
dz=z 2,  w  =  4,  /  =  7,  and  fife  16. 


SECTION    LI. 

Art.  1  l*%. 

1.  78  strokes. 

2.  7  =  65;  £=425 

3.  Z=70;  £=1360. 

4.  Z:=48£;  S=903f. 

5.  J=26f;  £=414f. 

6.  The  progression  is  20,  25,  30,  35,  40,  45,  50,  55. 


46  key.  LII,  LII1. 


7.  The  progression 

8.  The  progressior 
to  41  inclusive. 

i  is  6,  6f ,  74,  8,  8§,  9J,  10. 

i  is  5,  7,  9,  11,  &c,  all  the  odd  numbers 

9.    300  strokes. 

10.   950  trees. 

11.   31700  feet. 

12.  1608£. 

13.  8=ian\ 

14.  8  days. 

15.  5  days  and  10  days ;  55  miles  and  80  miles. 

16.   12,  17,  22. 

17.   4,6  and  8. 

18.    14,  12,  10,  8  and  6. 

19.   531. 

SECTION  LII. 

Art.  1  •51. 

1.    Performed.  . 

.n—i 

3.  a=J=T;S= 

4.  a^^-V: 

n— l n— l 

£> 7i—l n— I • 

\/l  —  y/tf 

'a-'.te-ij" 

sy-i(?-i) 

grn— 1     '  £n—  1 

Let  the  learner  verify  the  preceding  formulae,  by  making  a=.% 
q  —  Sf  w  =  3,  Z=  18,  and  £=26. 


SECTION    LIU 


Art.  152. 

1.  7=640;  £=1275. 

2.  z  =  ¥'T;  ^sr=iM- 


IJV  ****«  *7 

i 

3.  2,  10  and  50. 

4.  $4095. 

5.  Performed. 

6.  3,  15,  75,  375  and  1875. 

7.  7,  35  and  175. 

8.  8,  24  and  72. 

9.  Any  progression  in  which  the  ratio  if  %. 

10.  3,  5  and  7. 

11.  6. 

12.  10. 

13.  7  937+. 

14.  178  535+. 

15.  2. 

16.  6. 

17.  21. 

18.  91*. 

19.  Progression  2,  6,  18,  54,  162. 

20.  Progression  5,  30,  180,  1080. 

21.  Progression  3,  15*,  75,  375,  1875,  9375,  46875. 

22.  Progression  1,  2,  4,  8,  16,  32,  64,  128,  256,  512. 


SECTION    LIV. 

\rt.  153. 
1st  to  5th  inclusive,  performed.     ,g    „   or    */ jq 

6.  626.  '      '      ^ 

7.  10,  or -2.  19    25'or49« 

8.  4.  20.   ±8,  or  rhy/— 1331 

9.  f.  21.   729,  or  (—V-)6- 

11.   81.  22'   3,  or  ^/— 41. 

12th  to  15th  inclusive,  performed.  23.   2,  or  —  3. 

16.  2,  or—  3.  24.    25,  or  81. 

17.  zkv/^oriy/ITe.  25.    (£ ± /-t^' 


48  kbt.  LV 

26.   27,  or  (— «)* 

28.  81,  or  (—  J£)4. 

29.  4,  or  y. 

30.  4,  or  1. 


•( 


±v/62  +  4«c  — 6\* 


31. 

2a 

32d  and  33d  performed. 

34.  9,  or  — 12. 

35.  10,  or— 2. 

36.  l,or  lzkt/60. 

37.  6,  or  —  9. 

38.  9,  or  5  +  ^25. 

*>    -             .       dry/377  —  1 
rf9.   5,  or  —  o,  or  — E — 


Atl    Q             i       5  ±4/1329 
40.   3,  or  —  £,  or ^ . 


SECTION    LV. 

Art.  151. 

1.  Performed. 

2.  K==^,y  =  4ior«==2,3rs=5. 

3.  x=18,  y  =  3;  or  z=12£,  y~—  2£. 

4.  x  =  2,y  =  3;  or  i  =  -  46,y  =  15. 

5.  z  =  4,  y  =  2;  orz  =  —  4,  y  =  —  2. 
6th  to  11th  inclusive,  performed. 

12.  z  =  6,y=3;  or  x  =  —  6,  y  —  —  3. 

13.  a;=7,y=:3;  orx  =  3,y  =  7. 

14.  x  =  3,  y  =  5;   a?  =  —  3,  y  =  —  5;    x  =  5,  y  =  3 ;   or 
*  =  —  5,y  =  —  3. 


LV.  KEY.  49 

15.  z  =  9,  y  =  S;  or  x  =  —  9,  y  =  —  3. 

16.  zz=49,  y  =  25. 
17.'  z  =  625,  y  =  16. 

18.  z  =  5,  y  =  4;  or  x=. —  5,y  = —  4. 

19.  x  =  7,  y  =  3 ;  or  z  =  —  3,  y  =z  —  7. 

20.  z  =  1,  y  =  2;  x  =  —  1,  y  =  —  2 ;  z  :=  2,  y  =  1  ;    or 
x  =  —  2,y  =  — 1. 

21.  z  =  2,y  =  4;z=:4,y  =  2;z  =  3  +  v/21^=--3— \/21; 
or  z  =  3  —  ^/2l,  y  =  3  +  ^/21. 

22.  z  =  1,  y  =  2  ;  or  z  =  2,  y  =  1. 

23.  z  =  4,  y  =  9  ;  or  z  =  9,  y  =  4. 

24.  z  =  8,  y  =  2 ;  or  z  =  2,  y  =  8. 
25th  and  26th  performed. 

27.  z  =  20,  y=16. 

28.  z  =  5,  y  =  3. 

29.  z  z=  8,  y  =  6 ;  z  =  —  8,  y  =  —  6  ;  z  =  6,  y  =  8 ;  or 
z  =  —  6,  y  =  —  8. 

30.  z=14,  y  =  12. 

31.  z  =  27,  y  =  148£  ;  or  z  =  6,  y  =  33. 

32.  x  =  5,  y  =  ±3. 

33.  Performed. 

34.  z  =  3,y  =  l;  z  =  -.3?  y  =  — 1;  z  =  8  tf&  y  =  ^/J  ; 
orz  =  —  8^/i,  y  =  —  [/h 

35.  z  =  4,  y  =  3;    x  =  —  4,y==— -3;   a:  =  1 1 V ^7  y  = 
—  12\/£";  orz=  — ll\/^y  =  12V^ 

36.  x=l,y  =  2;  z  =  —  1,  y  =  —  2;  z  =  |,  y  =  —  V  > 
or  z  =  —  f,  y—  V- 

37.  z  =  4,y  =  2;  z  =  — 4,y  =  — 2;  iaB-^f=^j 


50 


KEY. 


LVI,  LVI] 


SECTION  LVI. 


Art.  155. 

1.  2048. 

2.  65536. 

3.  1048576. 

4.  4194304. 

5.  131072. 

6.  4194304. 

7.  32. 

8.  32. 

9.  16. 

10.  1024. 

11.  16. 

12.  64. 


13.  32768. 

14.  16384. 

15.  65536. 

16.  1048576. 

17.  1048576. 

18.  1048576. 

19.  32. 

20.  8. 

21.  8. 

22.  16. 

23.  8. 

24.  8. 


SECTION    LVII. 


Art.  162. 

1.  31048284. 

6. 

41041671. 

2.  4-7581015. 

7. 

0-4392806. 

3.  21048284. 

8. 

2-5228783. 

4.  34456042. 

9. 

2-6389892. 

5.  5-5152060. 

10. 

20467815 

Art.  163. 

1.  30. 

6. 

1-63472. 

2.  2000. 

7. 

0-3048. 

3.  3155888. 

8. 

00593562. 

4.  1309534. 

9. 

00016891. 

5.  221754-6. 

10. 

00002155. 

Art.  167. 

1.  1-8593785, 

or  9  8593785, 

3. 

4-5789370,  or 

6-5789370 

2.  3-7604225, 

or  77604225. 

4. 

T-8239087,  or 

9-8239087 

LVIII. 


KEY. 


51 


5. 

1-2179083, 

or  9-21 

79083. 

8. 

26596926. 

*>• 

0-7262335. 

9. 

2U628630. 

7. 

0-5682017. 

10. 

1-3322956,  or  93322956 

Art. 

168. 

1. 

0-273603. 

5. 

000003. 

2. 

00536312. 

6. 

0-772481. 

3. 

00002. 

7. 

00676142. 

4. 

000000004' 

H435. 

8. 

000274074. 

SECTION    LVIII, 


Art. 

169. 

1st  to  the  13th  inclusive, 

performed. 

14. 

151-807.     * 

33. 

3-48428. 

15. 

77190. 

34. 

0-422838. 

16. 

324-324. 

35. 

0-77992. 

17. 

00035991. 

36. 

0-99285. 

18. 

7-71707. 

37. 

0-867362. 

19. 

301182. 

38. 

071254. 

20. 

3581242. 

39. 

0-80818. 

21. 

4577-626. 

40. 

11 1204. 

22. 

499336. 

41. 

0-59905. 

23. 

5932-52. 

42. 

0-4809. 

24. 

47701902. 

43. 

172002. 

25. 

0187295. 

44. 

Performed. 

26. 

933333. 

45. 

23774. 

27. 

71197. 

46. 

11291. 

28. 

5554569. 

47. 

1-9699. 

29. 

768575. 

48. 

Performed. 

30. 

00000000243. 

49. 

705. 

31. 

165-3817. 

50. 

30-787. 

32. 

2-236068. 

51.    Progression  7,  86394,  10662,  1316,  16242,  20 


52  KEY.  LIX. 

Art.  170. 
1.   Performed. 

2       S — a log.  7 —  log,  a , 

*    q~W^~V  W  — log.  (£—«)  —  log.  (S—  /)"*" 

c  +  ^(g  —  1) log.(a  +  £g  —  .9^—  log<g 

?  log.  q 

4.  fl  =  7g-g(g-n;  >l  =  log-/-log'(^^±jg?  +  1 

log.  q  ~ 

5.  £=122S5;  ti=12. 

6.  q  —  3;  n  =  8. 

7.  Z=177147;  n  =  10. 

8.  a  =  2;  w  =  8. 


SECTION    LIX.      ' 

Art.  171. 

1st  to  5th  inclusive,  performed. 

6.  $569-801. 

7.  $194867. 

8.  $33324. 

9.  $1829-71. 

10.  $250. 

11.  14-137  years,  or  14  years,  1  month,  29  days 

12.  0-06085,  or  6TV  per  cent,  nearly. 

13.  11-8956  years,  or  11  years,  10  months,  22  days 

14.  18854  years,  or  18  years,  10  months,  7  days. 

15.  14-2067  years,  or  14  years,  2  months,  14  days 

16.  $644-781. 

17.  83310. 

18.  003205. 

19.  2691  years,  or  26  years,  10  months,  28  days. 
Art.  17S. 

1.   Performed.  2.    $3558-094 


LX.  KEF. 


SECTION    LX. 

Art.  173. 

1st  2d  and  3d  performed. 

,  4.    $7686-22. 
'  5.    $7857-98. 

6.  $1169  84. 

7.  19387  years,  or  19  years,  4  months,  19  days. 


53 


MISCELLANEOUS    QUESTIONS 

1.  A  $948,  B  $1896,  C  $2844,  D  $4740. 

2.  3  yards  at  9s. ;  5  yards  at  7s. 

3.  8  years. 

4.  $871|. 

5.  60  years. 

6.  55. 

7.  4  days. 

8.  20. 

9.  Breadth  30,  length  40  rods. 

10.  10  eagles ;  $30. 

11.  A  $60;  B  $48. 

12.  $1000. 

13.  $50. 

14.  ,£400,  .£500,  .£700. 

15.  800  men. 

16.  $20. 

17.  £280. 

18.  14  beggars ;  32s. 

19.  6£. 

20.  40,  20  and  12. 

21  Wheat  10s.,  rye  6s.  per  bushel 

22.  /T. 

23.  23. 

24  21  and  16. 
5* 


54  KM. 

25.  17  and  8. 

26.  24s. 

27.  53. 

28.  A  $20;  B  $50. 

29.  Greater  79;  less  35. 

30.  A  .£300;  B  £350. 

31.  Length  20,  breadth  18  feet 

32.  A$l;  B$l£;  C  $2. 

33.  15  and  12. 

34.  10  and  8. 

35.  12  and  19. 

36.  8  and  15. 

37.  40  sheep. 

38.  A  .£10 ;  B  .£20 ;  C  .£30  j  D  £40. 

39.  A  6,  B  5  miles  per  hour. 
10.  93. 

41.  8  hours. 

12.  6  hours,  or  31  hours. 

13.  3  and  243. 

44.  A  204,  B  200  acres. 

45.  5  and  4. 

46.  36  and  24. 

47  Breadth  100,  length  150  feet 

48.  12  and  10. 

49.  Breadth  40,  length  50  rods. 

50.  16  and  9. 

51.  6  and  4. 

52.  18  and  6. 

53.  3  and  5. 

54.  5  and  9. 

55.  81  scholars. 

56.  23. 

57.  Depth  of  1st  4  feet,  that  of  2d  5  feet. 

58.  8. 

59.  25. 

60  Court  30  by  19  feet ;  height  of  fence  4  feet 


KEY  55 

61.  12  yards. 

62.  150  miles. 

63.  25. 

64.  A  $200;  B  $300. 

65.  9  and  15. 

66.  741. 

67.  A  $81 ;  B  $41 ;  C  $21 ;  D  $11 ;  E  $6. 

Explanation. 

Let  v,  w,  x,  yt  and  z  represent  the  money  of  A,  B,  C,  D,  and 
E,  respectively,  when  they  began  to  play ;  and  let  S  represent 
the  known  amount  of  their  money,  which  is  5  .  32  or  $160. 

In  the  first  game  A  loses  the  whole  amount  minus  his  own 
share,  or  S — v  ;  he  will  therefore  have  left  v  —  (S — v),  or 
2  v  —  8.  After  this  his  money  is  doubled  four  times ;  he  will 
then  have  32  v  — \QS\  this  must  be  equal  to  $32  or  \S. 
Hence, 

32  v  — 16  S=  £ S.  from  which  we  have 

.  B,  in  the  first  game,  doubles  his  money ;  he  will  then  have 
2m>;  in  the  second  game  he  loses  S — 2  wy  and  will  have  left 
Aw  —  8.  Afterwards  his  money  is  doubled  three  times ;  he  will 
then  have  32  w  —  8  8.     Hence, 

32m>-8£  =  |#,  and^  =  i^z=$41. 
In  like  manner,  for  C's  money  we  find 

32  x  —  48=18,  and  1  =  ^=:  $21; 


for  D's  money, 
32 
and  for  E's  money, 


32y-2  S=  J  S,  and  y=  1^  =  111; 


32z-S=iS>andz  =  ^  = 


56 


KEY 


68. 

A  40,  B  60  eggs. 

69. 

10  and  14. 

70. 

4  and  2. 

71. 

4,  6,  and  11. 

72. 

Depths  4  feet  and  5  feet 

73. 

5  and  3. 

74. 

6. 

75. 

16. 

76. 

$50. 

77. 

7  and  3. 

78. 

7,  5  and  3. 

79. 

1,  3,  5,  ami  7. 

80. 

2,  6,  and  18. 

81. 

10,  20,  40,  and  80. 

82. 

.£94  18s.  Od.  1  qr. 

83. 

$491585. 

TJBI78E.SIT 


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